A229260 -id:A229260 - cp's OEIS frontend

A351117 a(n) = Sum_{k=0..n} k! * k^(kn) Stirling2(n,k).

1, 1, 33, 118483, 103098350565, 35763050750177408011, 7426387531294259002278007386693, 1294894837982331421844458945612619053737859003, 253092742000650212461957357208907985560332648454746968725711765

Offset: 0

Seiichi Manyama, Feb 01 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A000670, A122399, A229260, A242229.

Cf. A249584.

a:= n-> add(k!*k^(k*n)*Stirling2(n,k), k=0..n):
seq(a(n), n=0..10);  # Alois P. Heinz, Feb 01 2022

a[0] = 1; a[n_] := Sum[k! * k^(k*n) * StirlingS2[n, k], {k, 1, n}]; Array[a, 9, 0] (* Amiram Eldar, Feb 02 2022 *)

a(n) = sum(k=0, n, k!*k^(k*n)*stirling(n, k, 2));

my(N=10, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (exp(k^k*x)-1)^k)))

my(N=10, x='x+O('x^N)); Vec(sum(k=0, N, k!*(k^k*x)^k/prod(j=1, k, 1-k^k*j*x)))

E.g.f.: Sum_{k>=0} (exp(k^k*x) - 1)^k.
G.f.: Sum_{k>=0} k! * (k^k*x)^k/Product_{j=1..k} (1 - k^k*j*x).
a(n) ~ n! * n^(n^2). - Vaclav Kotesovec, Feb 04 2022

A227207 E.g.f.: Sum_{n>=0} n^n * x^n / (n! * Product_{k=0..n} (1 - nkx)).

Original entry on oeis.org

1, 1, 6, 105, 3568, 204745, 18028266, 2278860535, 394667414016, 90302033890953, 26525942216131330, 9775058594870836861, 4433256936788979640848, 2434899483389881601250937, 1597444746833206096334387802, 1237091666097626095124512681755, 1119205949224015886848972396596736

Offset: 0

Paul D. Hanna, Sep 18 2013

nonn

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 105*x^3/3! + 3568*x^4/4! + 204745*x^5/5! +...
where
A(x) = 1 + x/(1-x) + 2^2*x^2/(2!*(1-2*1*x)*(1-2*2*x)) + 3^3*x^3/(3!*(1-3*1*x)*(1-3*2*x)*(1-3*3*x)) + 4^4*x^4/(4!*(1-4*1*x)*(1-4*2*x)*(1-4*3*x)*(1-4*4*x)) +...

Seiichi Manyama, Table of n, a(n) for n = 0..212

Cf. A119392, A229257, A229258, A229259, A229260, A229261.

Cf. A229233, A229234, A220181, A122399.

Flatten[{1,Table[n! * Sum[k^n * StirlingS2[n,k] / k!,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 08 2014 *)

{a(n)=n!*polcoeff(sum(m=0, 20, m^m*x^m/m!/prod(k=1, m, 1-m*k*x +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=n!^2*polcoeff(sum(m=0, n, (exp(m*x+x*O(x^n))-1)^m/m!^2), n)}
for(n=0, 20, print1(a(n), ", "))

{a(n)=n!*sum(k=0, n, k^n*stirling(n, k, 2)/k!)}
for(n=0, 20, print1(a(n), ", "))

Sum_{n>=0} a(n)*x^n / n!^2 = Sum_{n>=0} (exp(n*x) - 1)^n / n!^2.

a(n) = n! * Sum_{k=0..n} k^n * Stirling2(n,k) / k!.

cp's OEIS Frontend

A351117 a(n) = Sum_{k=0..n} k! * k^(kn) Stirling2(n,k).

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A227207 E.g.f.: Sum_{n>=0} n^n * x^n / (n! * Product_{k=0..n} (1 - nkx)).

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cp's OEIS Frontend

A351117 a(n) = Sum_{k=0..n} k! * k^(k*n) * Stirling2(n,k).

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Author

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Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A227207 E.g.f.: Sum_{n>=0} n^n * x^n / (n! * Product_{k=0..n} (1 - n*k*x)).

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Author

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Examples

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Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A351117 a(n) = Sum_{k=0..n} k! * k^(kn) Stirling2(n,k).

A227207 E.g.f.: Sum_{n>=0} n^n * x^n / (n! * Product_{k=0..n} (1 - nkx)).